On the functional equation $$\hbox {f}(\hbox {x}+\hbox {y})=\hbox {g}(\hbox {xy})$$

Abstract

The functional equation $$f(x+y)=g(xy)$$ f ( x + y ) = g ( x y ) is investigated with unknown functions $$f:A+A\rightarrow Y$$ f : A + A → Y , $$g:A\cdot A\rightarrow Y$$ g : A · A → Y in the following cases: $$A:=\left]\alpha ,\beta \right[\subseteq \mathbb {F}_{+}$$ A : = α , β ⊆ F + where $$\mathbb {F}$$ F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set $$A\subseteq \mathbb {R}_{+}$$ A ⊆ R + that is closed under addition and multiplication and there exist functions f, $$g:A\rightarrow Y$$ g : A → Y which satisfy the equation $$f(x+y)=g(xy)$$ f ( x + y ) = g ( x y ) for all $$x,y\in A$$ x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given.